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SCIENCE { FUNCTION } ANALYSIS 



Man gewinnt dadurch schon sehr vie], wenn man eine 
Menge von Untersuchungen unter die Formel einer ein- 
zigen Aufgabe bringen kann. Denn dadurch erleich- 
tert man sich nicnt allein selbst sein eigenes Geschaft, 
indem man es sich genau bestimmt, sondern auch jedem 
anderen, der es priifen will, das Urtheil, ob wir unserem 
Vorhaben ein Geniige gethan haben oder nicht. — Kant. 



PROBLEM 

SCIENCE f FUNCTION } ANALYSIS 

FORMULA 

SCIENCE { } ANALYSIS 

SCIENCE { X J ANALYSIS 

INDIVIDUAL 

GENERAL , 

UNIVERSAL 

BY 
GEORGE ASHTON BLACK, PH.D. 



NEW YORK 

PRINTED FOR THE AUTHOR 

1905 






LIBRARY of CONGRESS 
I wo Copies Keceivea 

rEfl 2 IUU5 

UW.SS o AAC. NO! J 

copy e. 




Copyright, 1905, by 
George Ashton Black 



Published January, 1905 



The DeVinne Press. 






c^ 

^ 






SCIENCE { FUNCTION^ ANALYSIS 

SCIENCE { } ANALYSIS 
SCIENCE \X] ANALYSIS 

C INDIVIDUAL 1 
SCIENCE^ GENERAL ^ANALYSIS 
( UNIVERSAL J 



DEDUCTION OF 
PEOBLEM AND FOEMULA 

DEFINITION 

Science = necessary { cognition } universal = the reso- 
lution of all cases of a general problem = Analysis. 

Hence 
Science \ function \ Analysis 

Hence 

Scienc e { ] Analysis 

Science \ x \ Analysis 

( individual ^ 

Science < general > Analysis 

( universal ) 

which satisfies the definition whether by resolving the prob- 
lem into the singular case marked by the only imaginary 
differential coefficient \ } and the general case marked by 
any real differential coefficient { x } where x is individual 
general universal through all moments of the logical deter- 
mination 

( individual ^ 
real <. general > x 
( universal ) 

or by determining the problem in relation to all moments 
of a single necessary { cognition \ universal according to 
Kant's classic enumeration of them in their proper order 

7 



and connection ; for the only three fundamentally distinct 
moments of the formula differ whether as intuition concept 
idea according to the rule ideal— or as synthesis analysis 
dialectic according to the rule method— of Science { func- 
tion } Analysis. 

Thus the formula lays down the Science as well as the 
Analysis of my original problem in the shape of a series of 
derivative problems, the demanded resolution of which, in 
the order and connection prescribed by the formula, is de- 
veloped in the sequel into an organon, yet to be a complete 
system, of Science j universal } Analysis. 

Note. The isolated variety 



i * i 

individual 

general 

universal 



is necessary and sufficient to integrate or satisfy the isolated 
differential equation 



( intuition synthesis 

ideal < concept analysis )- method 

( idea dialectic 



to which Kant's enumeration leads through the recognition 
of the enumeration in two equivalent formulas. Witness my 



intuition \ \ synthesis 

concept \ x \ analysis 

ideal ( \ method 

( individual ) 

idea < general > dialectic 

( universal ) 



DEDUCTION OF 
WOEK DEMANDED BY FOEMULA 

POSTULATE 

I 

/ \ = a plane equilateral triangle the internal determination 
of which is only imaginary = the geometrical construction 
whether of the concept Science or of the concept Analysis 
referred to the vacant place of any correlative concept = 
the solution of one case of the general problem, Science 
\ } Analysis j 

/ \ = a plane equilateral triangle the internal determina- 
tion of which is real in respect of only the middle point 
of the altitude = the geometrical construction whether of the 
concept Science or of the concept Analysis referred to the 
concept individual = the solution of one case of the general 
problem, Science [individual } Analysis. 

Ill 

Resolution of all cases of Science { individual} Analysis = 
solution of one case of Science {general} Analysis. 

IV 

Resolution of all cases of Science {general} Analysis = solu- 
tion of one case of Science { universal } Analysis. 

HENCE 

the work by corresponding stages, as follows. 

9 



I Resolution of all cases of Science j \ Analysis achieved 
in and through the integration of a simple differential 
equation derived from the solution of one case. 



A 




of Science \ } Analysis 

synthesis 

analysis ) method 

dialectic 



intuition / \ synthesis 

concept • analysis 

ideal) I / \ \ me thod 

of Science \ I / I / of Analysis 

'idea /\ { / \ • dialectic' 



where the imaginary term, derived from the imaginary dif- 
ferential coefficient in 2, and posited in the shape of a blank 
space before Science and likewise before Analysis, should 
not be overlooked ; for it is necessary and sufficient in that 
place to indicate the singular case of my original problem, 
evidently the simplest case possible, which is alone in ques- 
tion now. 

10 



Note. 2 arises from distinction of successive stages in the 
reproduction of 1 by degrees according to a constant rule 
referred to Science { } Analysis in relation to which the 
whole of 1 was postulated. Thus the stages in question 

giveii Kant's terminology, are found to differ 



whether as intuition concept idea according to the rule ideal 
— or as synthesis analysis dialectic according to the rule 
method — of Science { } Analysis. 

3 arises from reference, in one to one correspondence, of the 
successive moments of two equivalent formulas obtained 
from 2 to the same whole of possible Science { \ Analysis, 
which is thereby perfectly differentiated a priori ; followed 
by the integration of the resulting differential equation. 
The second moment of the integral represents the necessary 
identity in all the variety presented by the first moment. 
For the first moment may be conceived to undergo a gradual 
shortening of unit line until all difference in the variety line 
angle surface vanishes, and nothing is left but the identity 
in the same, in the image of a point. So conceived the first 
moment stands to the second in the relation of variable to 
limit. The third moment of the integral represents the 
necessary reference of all the variety presented by the first 
moment, to the representation whether of the variable or 
limit in question. 



LofC. 



li 



II Resolution of all cases of Science {individuals Analysis 
achieved in and through the integration of a simple differ- 
ential equation derived from the solution of one case. 



/- intuition 
/• concept > ideal 

idea 

of Science \ individual] Analysis 

synthesis 

analysis ^ method 

dialectic 



intuition 



synthesis 



ideal 
of individual 
Science 



concept D analysis 

A B 



idea 



AB 

x 
CD dialectic 
II 
ABC-D 



method 

of individual 
Analysis 



12 



Ill Resolution of all cases of Science j general ] Analysis 
achieved in and through the integration of a complex dif- 
ferential equation derived from the solution of one case. 



C 

D 

A B 

AB 

x 
CD 

II 
ABC'D 



intuition 
concept ( 
idea 

synthesis 

analysis 

dialectic 



ideal 



of Science [ general \ Analysis 



method 



method of general Analysis 
synthesis analysis dialectic 



m 

CD 

bD 
o 



/ intuition 



concept 



idea 



C 

D 

A B 

AB 

x 

CD 

II 

ABC'D 



Z 

D 

X Y 



base multiplicand 
x x 

V% altitude multiplier 



area 



II 
product 



zZ 

D 

xX y Y 

base 

x 
i altitude = area 

II II II 

area = area 2 -f- area 



Yt. alt. 

x 
base 



Note. The complex form of the differential equation in 3 is 
rendered necessary by the specialty of the particular Science 
\ x } Analysis in question. Only through the cross refer- 
ence of the two equivalent formulas obtained from 2 could 
the same whole of possible Science { general } Analysis be 
perfectly differentiated a priori. 

13 



IV a Derivation from the solution of one case, of a com- 
pound differential equation demanding the resolution of all 
cases, of Science \ universal } Analysis. 



A 



c 

D 
A B 

AB 

x 

CD 

ABC'D 



Z 

D 

X Y 



base multiplicand 
x x 

% altitude multiplier 



area 



product 



zZ 

D 

xX yY 



= area 
x x 

Vz altitude = area 
II II 

area = area 2 



altitude 

x 
base 

II 
area 



single manifold restrictive intuition ^ 

definite definable definitive concept > ideal 

individual general universal idea ) 

2 of Science \ universal \ Analysis 

synthetical analytical dialectical synthesis 1 

" " " analysis > method 

" " " dialectic ) 



3 Here belong the empty tables ABC which follow. 
They perfectly differentiate the whole of possible Science 
5 universal ] Analysis, and constitute a compound differen- 
tial equation demanding the resolution of all cases of that 
problem. 



14 



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IV b Essay to refer recognized moments of the demanded 
resolution of all cases of Science { universal} Analysis each 
to its proper place in the formula demanding the resolution. 

On the supposition that pure reason, constant as the 
faculty of Science \ function } Analysis in all rationals of all 
times, but varied through all moments of a maximum dif- 
ferentiation and integration of that function in different 
rationals of different times, has somewhere in some context 
already cognized every step in the solution of every case of 
Science J universal} Analysis, but has not yet recognized 
every step in the solution of every case in its proper place in 
the resolution of all cases ; I propose to search out all and only 
the cognitions that are the content of that resolution, and 
arrange them each in that proper place as fixed for it a 
priori by Tables ABC. To be sure the task is not for 
only one rational, but for every one interested in the 
development as much as possible in himself of the same 
faculty that aforetime made the cognitions, and is now in 
his person called upon to recognize what it has itself in other 
persons already produced according to a fixed and ascertained 
formula. My own discovery of the required cognitions and 
reference of them to this or that place in the formula is sure 
and complete as regards the solution of only one case of the 
general problem in question j but all that is wanting to the 
perfection of the demanded resolution will undoubtedly be 
found, if able men, and such as are acquainted with what is 
classic in the use of pure reason, will endeavor to recognize 
the missing cognitions by the general but sufficient marks 
that relegate them to one or another place in the formula 
in correlation with one or another moment of the singular 
solution. 



18 



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II II II 
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